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Let $R_1$ and $R_2$ are two rings such that $R_1$ $\subseteq$ $R_2$ and $R_2$ is a finitely generated ring over $R_1$. We know that, if $R_1$ is Noetherian and $S$ is a ring such that $R_1$ $\subseteq$ $S$ $\subseteq$ $R_2$ and $R_2$ is a finitely generated module over $S$, then $S$ is a finitely generated ring over $R_1$.

I found out in the June edition of The American Mathematical monthly that if we remove the condition that "$R_2$ is a finitely generated module over $S$", then there exists examples where $S$ might not be a finitely generated ring over $R_1$ and they construct an example using heavy machineries, I was thinking whether a 'simpler' example could be constructed.

And Also if one could provide a general algorithm on constructing such rings, it will be very much appreciable.

Jyrki Lahtonen
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    I have no idea what it would mean to give a "general algorithm" here. – Eric Wofsey Jun 21 '17 at 04:27
  • @EricWofsey it means that my question has 2 parts, first part asks to give just an example of the fact and the second one asks to give a $general$ mechanism to construct such examples. –  Jun 21 '17 at 04:30
  • The first paragraph refers to the Artin-Tate lemma. – user49640 Jun 21 '17 at 04:39
  • I am sure examples of this have been given in this site a few times — I myself have written a couple of answers, in fact! Have you looked before asking? – Mariano Suárez-Álvarez Jun 21 '17 at 05:24
  • Tip: wrapping a text with asterisks produces italics. Using double asterisks gives bold. The spacing in such italics is better than what you get with $math$ $italics$, where letters are treated as symbols (as they should). If you prefer the latter, then I apologize. We can roll my edit back. – Jyrki Lahtonen Jun 21 '17 at 06:00

1 Answers1

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Let $R_1 = k, R_2 = k[x,y], S = k[x,xy,xy^2,xy^3,\dots]$.

user49640
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